UDC 517.946

MATHEMATICS

V. G. MAZ’YA, V. P. KHAVIN

A NONLINEAR ANALOGUE OF THE NEWTONIAN POTENTIAL AND METRIC PROPERTIES OF \((p,l)\)-CAPACITY

(Presented by Academician L. V. Kantorovich, 20 III 1970)

In many questions of analysis, the massiveness of a set \(E\) located in Euclidean space \(R^n\) is naturally characterized by means of the quantity
\[ \inf \{\|\nabla_l\varphi\|_p^p:\varphi\in \mathcal U(E)\}, \]
where \(\|\ \|_p\) is the norm in \(L_p=L_p(R^n)\), \(\mathcal U(E)=\{\varphi\in\mathcal D:\varphi(x)\geq 1\ \text{for any } x\in E\}\), \(\mathcal D\) is the set of all functions of class \(C^\infty(R^n)\) with compact supports, and \(\nabla_l\varphi\) is the gradient of order \(l\) of the function \(\varphi\) (see, for example, \((^{1-14})\)). This quantity, called the \((p,l)\)-capacity of the set \(E\) and denoted below by \(\operatorname{cap}_{p,l}(E)\), for \(p=2,\ l=1\) and \(n\geq 3\) coincides with the classical capacity studied in potential theory (see \((^{15-17})\)).

In this paper an extension of the Newtonian potential is proposed which, in our opinion, is well adapted to the study of the function \(\operatorname{cap}_{p,l}\). With the help of this extension it has been possible to find metric conditions necessary and (separately) sufficient for the vanishing of \((p,l)\)-capacity (see below, item 5). For \(p=2\) these conditions coincide with the known conditions of the potential theory of M. Riesz (see \((^{15})\), Ch. III, Theorems 3.13 and 3.14; \((^{16})\), § IV). The theorems of item 5 refine some results of the papers \((^{6,7,14})\).

Throughout what follows we shall assume that \(p>1\), \(l\) is a positive (not necessarily integral) number, and that \(pl<n\).

1. The function \(c_{p,l}\) and the space \(\mathscr L_p^l\). For a function \(\varphi\) of class \(\mathcal D\) put
\[ \Lambda_{x\to y}^{\,l}\varphi=F_{\xi\to y}^{-1}\bigl(|\xi|^l F_{x\to \xi}(\varphi)\bigr), \]
where \(F\) is the Fourier transform. We introduce in \(\mathcal D\) the norm
\[ \|\varphi\|_{p,l}=\|\Lambda^l\varphi\|_p. \]
We define on the system of all compact subsets of the space \(R^n\) the capacity \(c_{p,l}\):
\[ K \xrightarrow{\;c_{p,l}\;} \inf\{\|\varphi\|_{p,l}^p:\varphi\in\mathcal U(K)\}. \]

Let us note that the ratio of \(c_{p,l}(K)\) to \(\operatorname{cap}_{p,l}(K)\) is bounded above and separated from zero uniformly with respect to all compact sets \(K\subset R^n\), since for integral \(l\) the norm \(\|\ \|_{p,l}\) on \(\mathcal D\) is equivalent to the norm \(\varphi\mapsto\|\nabla_l\varphi\|_p\). Starting from the set function \(c_{p,l}\), we define in the usual way the inner and outer capacities \(\underline c_{p,l}(E)\) and \(\overline c_{p,l}(E)\) for any subset \(E\) of the space \(R^n\) (see \((^{15-18})\)). If \(E\) is an analytic (and, in particular, Borel) set, then
\[ \overline c_{p,l}(E)=\underline c_{p,l}(E). \]
In this case, instead of \(\overline c_{p,l}(E)\) we shall write \(c_{p,l}(E)\).

We denote by \(\mathscr L_p^l\) the completion of the space \(\mathcal D\) with respect to the norm \(\|\ \|_{p,l}\). This space can be embedded naturally in \(L_s\), where
\[ s=np(n-pl)^{-1}. \]
We shall assume that the operator \(\Lambda^l:\mathcal D\to L_p\) has been extended by continuity to the whole space \(\mathscr L_p^l\).

Lemma 1. The operator \(\Lambda^l\) maps \(\mathscr L_p^l\) isometrically onto \(L_p\). The operator inverse to \(\Lambda^l\) (denoted by \(\Lambda^{-l}\)) is given by the equality
\[ (\Lambda^{-l}f)(x)=c\int \frac{f(y)}{|x-y|^{n-l}}\,dy \quad (x\in R^n) \]
(\(c\) is a constant depending only on \(n,p,l\); the integral is taken over \(R^n\)).

2. Energy

It is easy to see that every functional \(T \in (\mathscr L_p^l)^*\) can be identified with some generalized function in the sense of Schwartz. The energy of a generalized function \(T \in (\mathscr L_p^l)^*\) will mean the number

\[ \mathscr E_{p,l}(T)\stackrel{\mathrm{def}}{=}\|T\|_{(\mathscr L_p^l)^*}^{q},\quad \text{where } q=p(p-1)^{-1}. \]

By the word measure we shall everywhere below mean a nonnegative countably additive locally finite function, defined on the Borel \(\sigma\)-ring of the space \(R^n\). Identifying the functional \(\varphi\to \int \varphi\,d\mu\) \((\varphi\in \mathscr D)\) with the measure \(\mu\), we can formulate the following result.

Lemma 2. A measure \(\mu\) is a generalized function with finite \((p,l)\)-energy if and only if

\[ \int\left[\int \frac{d\mu(y)}{|x-y|^{\,n-l}}\right]^q dx<+\infty \quad \left(q=\frac{p}{p-1}\right). \]

The set of all such measures (denoted below by \(\mathfrak M_{p,l}\)) is closed in the space \((\mathscr L_p^l)^*\).

The last assertion, for \(p=2\) and \(l=1\) \((n\geqslant 3)\), turns into the well-known theorem of A. Cartan (see \((^{15})\), p. 117, Theorem 1.18).

3. Potentials

Let \(T\in(\mathscr L_p^l)^*\). Put

\[ U_{p,l}^{T}=\Lambda^{-l}\left(\left|(\Lambda^{-l})^*T\right|^{(2-p)/(p-1)}(\Lambda^{-l})^*T\right) \]

(the asterisk denotes passage to the adjoint operator). It is easy to see that \(U_{p,l}^{T}\in \mathscr L_p^l\). The function \(U_{p,l}^{T}\) will be called the \((p,l)\)-potential of the generalized function \(T\). Note that for \(p=2\) we return to the well-known definition of the Riesz potential of a generalized function \((^{15})\), p. 434). If \(\mu\in\mathfrak M_{p,l}\), then for almost all \(x\in R^n\)

\[ U_{p,l}^{\mu}(x)=c\int\left[\int \frac{d\mu(z)}{|z-y|^{\,n-l}}\right]^{1/(p-1)} \frac{dy}{|y-x|^{\,n-l}}. \]

Let us also record the expression for the energy of a measure \(\mu\in\mathfrak M_{p,l}\):

\[ \mathscr E_{p,l}(\mu)=\int U_{p,l}^{\mu}\,d\mu. \]

Lemma 3. The image of the space \((\mathscr L_p^l)^*\) under the mapping \(T\to U_{p,l}^{T}\) is equal to \(\mathscr L_p^l\).

Theorem 1 (rough maximum principle). Let the measure \(\mu\) be such that \(U_{p,l}^{\mu}(x)\leq M\) for every \(x\) in the (closed) support of the measure \(\mu\). Then for every \(x\in R^n\) the inequality \(U_{p,l}^{\mu}(x)\leq cM\) holds, where \(c\) depends only on \(n,p\), and \(l\).

Theorem 2. Let \(\mu\) be a measure with compact support. If \(U_{p,l}^{\mu}(x)<+\infty\) for \(\mu\)-almost all \(x\in R^n\), then for every \(\varepsilon>0\) one can indicate such a compact set \(K\subset R^n\) that the \((p,l)\)-potential of the restriction of the measure \(\mu\) to the set \(K\) is continuous in \(R^n\) and \(\mu(R^n\setminus K)<\varepsilon\).

4. Capacitary distribution

Let \(K\) be a compact set in \(R^n\), and let \(\{\varphi_m\}\) be some sequence minimizing the norm of the space \(\mathscr L_p^l\) on the set \(\mathcal U(K)\):

\[ \lim \|\varphi_m\|_{p,l}^{p}=C_{p,l}(K),\quad \varphi_m\in\mathcal U(K),\quad \text{for all } m. \]

From the uniform convexity of the space \(\mathscr L_p^l\) it follows that this sequence converges in \(\mathscr L_p^l\) to a certain function \(V_K\in\mathscr L_p^l\), which does not depend on the choice of the sequence \(\{\varphi_m\}\).

Theorem 3. The function \(V_K\) is the \((p,l)\)-potential of some measure \(\mu_K\in\mathfrak M_{p,l}\), concentrated on \(K\), and such that:

1) \(U_{p,l}^{\mu_K}(x)\geqslant 1\) for every \(x\in K\), except for a set of points of zero capacity \(c_{p,l}\);

2) \(U_{p,l}^{\mu_K}(x)\leqslant 1\) for every \(x\in \operatorname{supp}\mu_K\);

3) \(\mu_K(K)=\displaystyle\int U_{p,l}^{\mu_K}\,d\mu_K=c_{p,l}(K)\).

From this theorem it is not hard to derive that the measure \(\mu_K\) and the capacity \(c_{p,l}(K)\) are solutions of certain extremal problems analogous to the classical ones \(({}^{15})\), definition on p. 169, the definition of Vallée-Poussin on p. 176). It can be shown that, analogously to the classical theory of potential, for every bounded subset \(E\) of the space \(R^n\) there exist “outer” and “inner” capacitary distributions, which coincide with one another if \(E\) is an analytic set (see \(({}^{15})\), Ch. II, Theorems 2.6 and 2.7).

5. Metric characteristics of sets of zero \((p,l)\)-capacity. Denote by \(m_h(E)\) the Hausdorff \(h\)-measure of the set \(E\) (see \(({}^{15})\), pp. 244–245). If \(h(t)=t^\alpha\), then instead of \(m_h(E)\) we shall write \(m_\alpha(E)\).

Theorem 4. Suppose that

\[ \int_0^1\left(\frac{h(s)}{s^{\,n-lp}}\right)^{1/(p-1)}\frac{ds}{s}<+\infty. \]

If \(E\) is a Borel set \((E\subset R^n)\) for which \(m_h(E)>0\), then \(c_{p,l}(E)>0\).

For the proof of this theorem we verify that the potential of a measure concentrated on \(E\) is bounded (cf. \(({}^{16})\), p. 29).

Theorem 5. If \(m_{n-lp}(E)<+\infty\) (\(E\) is a Borel set in \(R^n\)), then \(c_{p,l}(E)=0\).

Consider now the \(n\)-dimensional Cantor set \(e\), equal to the intersection of a decreasing sequence of sets \(\{e_k\}_{k=1}^{\infty}\), where \(e_k\) is equal to the sum of \(2^{kn}\) closed cubes with side \(l_k\).

Theorem 6. The condition

\[ \sum_{k=1}^{\infty}2^{-kn/(p-1)}l_k^{-(n-pl)/(p-1)}=+\infty \]

is necessary and sufficient in order that the set \(e\) have zero capacity \(c_{p,l}\).

This theorem generalizes Otsuka’s theorem (see \(({}^{13})\), p. 31). In the proof of Theorem 7 the following lemma is used, which generalizes a theorem of Nevanlinna (see \(({}^{16})\), p. 30) and is based on an estimate of the \((p,l)\)-potentials of certain measures.

Lemma 4. If for every \(r>0\) the Borel set \(E\subset R^n\) can be covered by \(A(r)\) closed balls of radii not exceeding \(r\), and if

\[ \int_0 [A(r)r^{\,n-lp}]^{-1/(p-1)}\frac{dr}{r}=+\infty, \]

then \(c_{p,l}(E)=0\).

6. Removal of singularities of analytic functions. Let \(D\) be a domain on the extended complex plane, containing \(\infty\), and let \(H_p(D)\) be the set of all functions holomorphic in \(D\) and belonging to \(L_p(D)\). It is not difficult to verify that \(H_p(D)\) consists only of the zero function if and only if \(c_{q,1}(E)=0\). Therefore the following assertion is valid, supplementing a theorem of Carleson \(({}^{16})\), p. 73):

Theorem 7. Let \(p>2\). The set \(H_p(D)\) contains only the zero function if there exists a function \(h\) such that \(m_h(R^2\setminus D)>0\), and

\[ \int_0^1 \left[\frac{h(t)}{t}\right]^{p-1} dt<+\infty . \]

In a similar way one can sharpen certain known theorems on the removability of singularities of solutions of elliptic equations, formulated in terms of Hausdorff measures (see \((^5,^7,^16)\)).

Leningrad State University
named after A. A. Zhdanov

Received
6 I 1970

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